Dade Groups for Finite Groups and Dimension Functions

Let \(G\) be a finite group and \(k\) an algebraically closed field of characteristic \(p>0\). We define the notion of a Dade \(kG\)-module as a generalization of endo-permutation modules for \(p\)-groups. We show that under a suitable equivalence relation, the set of equivalence classes of Dade \(kG\)-modules forms a group under tensor product, and the group obtained this way is isomorphic to the Dade group \(D(G)\) defined by Lassueur. We also consider the subgroup \(D^{\Omega} (G)\) of \(D(G)\) generated by relative syzygies \(\Omega_X\), where \(X\) is a finite \(G\)-set. If \(C(G,p)\) denotes the group of superclass functions defined on the \(p\)-subgroups of \(G\), there are natural generators \(\omega_X\) of \(C(G,p)\), and we prove the existence of a well-defined group homomorphism \(\Psi_G:C(G,p)\to D^\Omega(G)\) that sends \(\omega_X\) to \(\Omega_X\). The main theorem of the paper is the verification that the subgroup of \(C(G,p)\) consisting of the dimension functions of \(k\)-orientable real representations of \(G\) lies in the kernel of \(\Psi_G\).